Base Converter

What the base converter does

This converts a number from one base to another, and it is not limited to the usual suspects. You can go from any base between 2 and 36 to any other, so binary to base 7, base 13 to hexadecimal, whatever you need. Enter the number, choose where it is coming from and where it is going, and read off the result.

It also copes with very large numbers without losing precision, so it is just as happy with a long value as a short one.

How to use it

1. Enter your number. Type it using the digits valid for its current base.
2. Choose the From base and the To base. Pick the base the number is in now, and the base you want it in.
3. Press Calculate. The converted number appears in the result.

Press Reset to clear everything.

What a base really is

A base is simply how many different digits a number system counts with before it rolls over into a new column. Base 10, the everyday one, uses ten digits, 0 through 9, and each column is worth ten times the column to its right. Base 2 uses two digits and each column is worth twice the last. The pattern is always the same: in base b, the columns are worth the powers of b, so ones, then b, then b times b, and on up. Change the base and you only change that one number, b, that everything is built from.

How the conversion works

Converting between two bases is easiest with a common meeting point in the middle, and that point is the number's plain value. First the tool reads your number in its source base, multiplying each digit by its column's worth and adding them up, which gives the plain quantity it represents. Then it writes that quantity out in the target base by repeatedly dividing and collecting remainders. Read the number in, write it back out in the new base. Every base-to-base conversion is those two halves joined together.

A worked example with real numbers

Let us convert 412 from base 5 to base 16. First read it in base 5, where the columns are worth 25, 5, and 1:

• 4 × 25 = 100
• 1 × 5 = 5
• 2 × 1 = 2
• Total: 100 + 5 + 2 = 107

Now write 107 in base 16. Dividing, 107 is 6 sixteens with 11 left over, and 11 is the hex digit B. So 107 becomes 6B. That makes 412 in base 5 equal to 6B in hexadecimal, the same quantity dressed in two different bases.

Digits beyond nine

Once a base passes 10, it needs more than ten single digits, and there are not enough number symbols to go round. So letters step in. The letter a stands for 10, b for 11, and so on up to z for 35, which is why base 16 uses a through f and the largest base here, base 36, uses the full 0 through 9 and a through z. They are not letters in any real sense, just digits that ran out of numerals, each one standing for a single value.

Questions people ask

Which bases can it convert between?

Any base from 2 to 36, in either direction. You choose the base the number starts in and the base you want it converted to.

How does base conversion work?

The number is first read in its starting base to find its plain value, then written out in the target base by repeated division. The plain value is the common point both bases pass through.

Why do some results contain letters?

Bases above 10 need more than ten digits, so letters fill in: a is 10, b is 11, up to z for 35. Each letter is a single digit standing for that value.

Can it handle very large numbers?

Yes. It converts large values without losing precision, so long numbers come out exactly, not rounded.

References

1. Weisstein, Eric W. "Base." From MathWorld, A Wolfram Resource. https://mathworld.wolfram.com/Base.html